基于单元子分法的结构多尺度边界单元法

建立在基于单元子分法的一种有效自适应格式以及多区域边界元三步求解技术基础上提出了一种计算结构多尺度问题的多区域边界元法。首先,通过高斯积分误差分析公式确定边界单元在满足精度要求下所需要的高斯点数,当所需高斯点数超过规定数目时该单元就被自动划分成一定数量的子单元,从而消除结构多尺度所引起的近奇异性。在单元子分技术的基础上采用多区域边界元三步求解技术来处理材料非均质问题:第一步消除各子域的内部未知量,第二步消除各子域独自拥有的边界未知量,第三步根据位移相容性条件和面力平衡条件建立系统方程组并求解公共界面节点位移以及每个子域的其他未知量。数值算例结果表明本方法可以用较少的计算时间得到满意的结果,是处理结构多尺度问题的一种有效方法。     

An eight noded composite plate element for the dynamic analysis of spatial mechanism systems

The development of a shear-deformable laminated plate element, based on the Mindlin plate theory, for use in large reference displacement analysis is presented. The element is sufficiently general to accept an arbitrary number of layers and an arbitrary number of orthotrophic material property sets. Coordinate mapping is utilized so that non-rectangular elements may be modeled. The Gauss quadrature method of numerical integration is utilized to evaluate volume integrals. A comparative study is done on the use of full Gauss quadrature, reduced Gauss quadrature, mixed Gauss quadrature, and closed form integration techniques for the element. Dynamic analysis is performed on the RSSR (Revolute-Spherical-Spherical-Revolute) mechanism, with the coupler modeled as a flexible plate. The results indicate the differences in the dynamic response of the transverse shear deformable eight-noded element as compared to a four-noded plate element. Dynamically induced stresses are examined, with the results indicating that the primary deformation mode of the eight-noded Mindlin plate model being bending.     

基于核重构思想的最小二乘配点型无网格方法

介绍重构核点法的基本原理和近似函数的构造方法，并基于核重构思想，应用配点法和最小二乘原理，离散微分方程，建立求解的代数方程，提出了一种基于核重构思想的最小二乘配点型无网格方法．与一般配点法相比，该方法的系数矩阵是有对称正定的，计算精度高，稳定性好．该方法的实施不需要背景网格，不需要进行高斯积分，与Galerkin法相比，具有计算量小、边界条件处理简单的特点，是一种真正的无网格法．对该方法构造过程中的近似函数及其导数的计算、修正函数的计算及方法的实现等问题进行了探讨．文中结合若干典型算例，检验了该方法的有效性．     

Shape reconstruction of an inverse boundary value problem of two‐dimensional Navier–Stokes equations

This paper is concerned with the problem of the shape reconstruction of two‐dimensional flows governed by the Navier–Stokes equations. Our objective is to derive a regularized Gauss–Newton method using the corresponding operator equation in which the unknown is the geometric domain. The theoretical foundation for the Gauss–Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the boundary curve in the sense of a domain derivative. The numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. Copyright © 2009 John Wiley & Sons, Ltd.     

基于对偶变量变分原理和两端位移独立变量的保辛方法

将广义位移和动量同时用拉格朗日多项式近似,并选择积分区间两端位移为独立变量,然后基于对偶变量变分原理导出了哈密顿系统的离散正则变换和对应的数值积分保辛算法。当位移和动量的拉格朗日多项式近似阶数满足一定条件时,可以自然导出保辛算法的不动点格式。通过数值算例分析了位移和动量采用不同阶次插值所需最少Gauss积分点个数,并讨论了位移插值阶数、动量插值阶数以及Gauss积分点个数对保辛算法精度的影响,说明了上述不动点格式恰好是一种最优格式。     

多体系统动力学的高斯拘束分析方法

为研究高斯拘束原理下多体系统的动力学分析问题,针对一类多体开环链状结构,运用高斯拘束方法建立了动力学方程,讨论了动力学方程的符号推导过程,并给出了封闭形式的动力学方程解析表达式.以刚性和柔性结构为例,比较分析了不同分析方法、不同自由度下符号推导多体结构动力学方程的运算时间.分析结果表明,高斯拘束方法与传统的拉格朗日方法相比,更适合于多体结构动力学方程的符号推导,且结构自由度越高,其运算优势越明显.高斯拘束方法为一种较好的多体系统动力学分析方法.     

自由下落猫姿态最优控制的混合优化策略

研究猫自由下落时的转体运动对探索宇航员在太空失重状态下的空间运动规律具有重要的参考价值.针对猫自由下落时四肢先着地现象的姿态最优控制问题,提出一种由Gauss伪谱法求解可行解与直接打靶法求解最优精确解相结合的混合优化策略.首先,根据猫在自由下落过程中的角动量守恒原理,推导出简化后的两对称刚体系统的非完整姿态运动方程;然后基于Gauss伪谱法将此无漂移系统的姿态非完整运动规划问题离散为非线性规划问题,并在不考虑实际性能指标函数的条件下利用序列二次规划算法求解此非线性规划问题在较少节点时对应的控制变量可行解,再通过三次样条插值获取较多节点时的控制变量值;最后基于直接打靶法将插值得到的控制值作为序列二次规划算法的初始猜测值,从而求解得到最优的控制输入,再代入系统姿态运动方程,通过数值积分得到落体猫的转体姿态运动曲线.通过数值仿真,求解得到的姿态运动曲线是光滑的,能以较高的精度到达预定的目标姿态;最优控制输入也能满足预先设计的零边界控制要求以及最大控制要求;结果表明了此混合优化策略具有较强的鲁棒性和有效性.     

A STUDY ON THE WEIGHT FUNCTION OF THE MOVING LEAST SQUARE APPROXIMATION IN THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless method is a new numerical technique presented in recent years .It uses the moving least square (MLS) approximation as a shape function . The smoothness of the MLS approximation is determined by that of the basic function and of the weight function, and is mainly determined by that of the weight function. Therefore, the weight function greatly affects the accuracy of results obtained. Different kinds of weight functions, such as the spline function, the Gauss function and so on, are proposed recently by many researchers. In the present work, the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method. The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed. Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and a in Gauss and exponential weight functions are in the range of reasonable values, respectively, and the higher the smoothness of the weight function, the better the features of the solutions.     

薄板分析的线性基梯度光滑伽辽金无网格法

薄板问题的控制方程为四阶微分方程,因而当采用伽辽金法进行分析时,形函数需要满足C$^{1}$连续性要求,且至少使用二次基函数才能保证方法的收敛性.无网格形函数虽然易于满足C$^{1}$连续性要求,但由于不是多项式,其二阶导数的计算较为复杂耗时,同时也对刚度矩阵的数值积分提出了更高的要求.本文提出了一种薄板分析的线性基梯度光滑伽辽金无网格法,该方法的基础是线性基无网格形函数的光滑梯度.在梯度光滑构造的理论框架内,无网格形函数的二阶光滑梯度可以表示为形函数一阶梯度的线性组合,因而可以提高形函数二阶梯度的计算效率.分析表明,线性基无网格形函数的光滑梯度不仅满足其固有的线性梯度一致性条件,还满足本属于二次基函数对应的额外高阶一致性条件,因此能够恰当地运用到薄板结构的伽辽金分析.此外,插值误差分析也很好地验证了线性基无网格光滑梯度的收敛特性.算例结果进一步表明,线性基梯度光滑伽辽金无网格法的收敛率与传统二次基伽辽金无网格法相当,但精度更高,同时刚度矩阵所需的高斯积分点数明显减少.      

饱和两相介质动力固结问题时域求解的精细时程积分方法

针对u-p形式的饱和两相介质波动方程,采用精细时程积分方法计算固相位移u,采用向后差分算法求解流体压力p,建立了饱和两相介质动力固结问题时域求解的精细时程积分方法。针对标准算例,对该方法的计算精度进行了校核。开展了该方法相关算法特性的研究,对采用不同数值积分方法计算非齐次波动方程特解项计算精度的差异进行了对比研究,并对采用不同积分点数目的高斯积分法计算特解项条件下计算精度的差异进行了对比研究。研究结果表明,(1)该方法具有良好的计算精度。(2)计算非齐次波动方程特解项的数值积分方法中,梯形积分法的计算精度最差,高斯积分法、辛普生积分法和科茨积分法都具有较好的计算精度。(3)增加高斯积分点数目对于提高计算精度的作用并不显著。     

基于Gauss伪谱法的3D刚体摆姿态最优控制

研究Gauss伪谱法求解3D刚体摆姿态最优控制问题．针对其最优姿态控制问题,既要满足由任意位置运动到平衡位置姿态运动规划问题,又要满足系统含有动力学约束的力学模型问题,提出基于四元数来描述3D刚体摆的数学模型,建立3D刚体摆姿态的动力学和运动学方程,为了解决3D刚体摆在平衡位置处的姿态最优控制问题,设计基于Gauss伪谱算法的最优姿态开环控制器,得到了3D刚体摆的姿态最优控制轨迹,得到满足的可行解,通过仿真实验验证了其开环解在平衡位置的控制姿态最优性．     

Two-level stabilized finite element method for Stokes eigenvalue problem

A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered.This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh size H and a Stokes problem on a fine mesh with mesh size h = O(H 2),which can still maintain the asymptotically optimal accuracy.It provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution,which involves solving a Stokes eigenvalue problem on a fine mesh with mesh size h.Hence,the two-level stabilized finite element method can save a large amount of computational time.Moreover,numerical tests confirm the theoretical results of the present method.     

A PREDICTION TECHNIQUE FOR DYNAMIC ANALYSIS OF FLAT PLATES IN MID-FREQUENCY RANGE

The wave-based method (WBM) has been applied for the prediction of mid-frequency vibrations of flat plates.The scaling factors,Gauss point selection rule and truncation rule are introduced to insure the wave model to converge.Numerical results show that the prediction tech- nique based on WBM is with higher accuracy and smaller computational effort than the one on FEM,which implies that this new technique on WBM can be applied to higher-frequency range.     

变加速动力学系统的广义高斯最小拘束原理

变加速运动在日常生活和工程问题中普遍存在.变加速动力学又称牛顿猝变动力学,因其在混沌理论和非线性动力学中的应用而获得广泛关注.高斯原理是一个具有极值性质的微分变分原理.因此,研究变加速动力学系统的广义高斯原理在理论和应用两方面都有重要意义.文章提出并研究变加速动力学系统的广义高斯原理.首先,引入急动度空间的广义高斯变分概念,将质点的达朗贝尔原理对时间求导数后与广义高斯变分点乘,并利用高斯意义下的理想约束条件,建立了变加速动力学系统的广义高斯原理.在此基础上,通过构造广义拘束函数建立并证明变加速动力学系统的广义高斯最小拘束原理,并给出原理的阿佩尔形式、拉格朗日形式和尼尔森形式.其次,研究原理对变质量力学的推广.从密歇尔斯基方程出发,将它对时间求导并与广义高斯变分点乘,建立了具有理想约束的变质量变加速动力学系统的广义高斯原理.通过构造变质量系统的广义拘束函数,建立并证明变质量力学系统变加速运动的广义高斯最小拘束原理.文中以开普勒-牛顿空间问题为例,利用所得的广义高斯最小拘束原理方法进行计算,验证了方法的有效性.     

非均质中厚板的无网格LRPIM动力学分析

用局部加权残值法建立了非均质中厚板的局部径向点插值离散系统方程,采用无网格局部径向点插值法分析了非均质中厚板的自由振动和强迫振动问题。用径向基函数耦合多项式基函数来近似试函数,用四次样条函数做为加权残值法中的权函数。所构造的形函数具有Kronecker delta性质,可以很方便地施加本质边界条件。该方法不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行。在计算过程中,取积分中的高斯点的材料参数来模拟问题域材料特性的变化。计算结果表明,利用该方法计算非均质中厚板的自由振动和强迫振动问题可以得到具有较高精度的解。     

Partitioned subiterative coupling schemes for aeroelasticity using combined interface boundary condition method

The combined interface boundary condition (CIBC) method has been recently proposed for fluid–structure interaction. The CIBC method employs a Gauss–Seidel-like procedure to transform traditional interface conditions into velocity and traction corrections whose effect is controlled by a dimensional parameter. However, the original CIBC method has to invoke the uncorrected traction when forming the traction correction. This process limits its application to fluid–rigid body interaction. To repair this drawback, a new formulation of the CIBC method has been developed by using a new coupling parameter. The reconstruction is simple and the structural traction is removed completely. Two partitioned subiterative coupling versions of the CIBC method are developed. The first scheme is an implicit strategy while the second one is a semi-implicit strategy. Iterative loops are actualised by the fixed-point algorithm with Aitken accelerator. The obtained results agree with the well-documented data, and some famous flow phenomena have been successfully detected.     

A non‐iterative implicit algorithm for the solution of advection–diffusion equation on a sphere

A numerical algorithm for the solution of advection–diffusion equation on the surface of a sphere is suggested. The velocity field on a sphere is assumed to be known and non‐divergent. The discretization of advection–diffusion equation in space is carried out with the help of the finite volume method, and the Gauss theorem is applied to each grid cell. For the discretization in time, the symmetrized double‐cycle componentwise splitting method and the Crank–Nicolson scheme are used. The numerical scheme is of second order approximation in space and time, correctly describes the balance of mass of substance in the forced and dissipative discrete system and is unconditionally stable. In the absence of external forcing and dissipation, the total mass and L₂‐norm of solution of discrete system is conserved in time. The one‐dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman–Morrison's formula and Thomas's algorithm. The one‐dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The suggested method is direct (without iterations) and rapid in realization. It can also be applied to linear and nonlinear diffusion problems, some elliptic problems and adjoint advection–diffusion problems on a sphere. Copyright © 2015 John Wiley & Sons, Ltd.     

结构动力方程的更新精细积分方法

将高斯积分方法与精细积分方法中的指数矩阵运算技巧结合起来,建立了精细积分法的更新形式及计算过程,对该更新精细积分方法的稳定性进行了论证与探讨。在实施精细积分过程中不必进行矩阵求逆,整个积分方法的精度取决于所选高斯积分点的数量。这种方法理论上可实现任意高精度,计算效率较高,其稳定性条件极易满足。数值例题也显示了这种方法的有效性。     

Three‐dimensional transient Navier–Stokes solvers in cylindrical coordinate system based on a spectral collocation method using explicit treatment of the pressure

A spectral collocation method is developed for solving the three‐dimensional transient Navier–Stokes equations in cylindrical coordinate system. The Chebyshev–Fourier spectral collocation method is used for spatial approximation. A second‐order semi‐implicit scheme with explicit treatment of the pressure and implicit treatment of the viscous term is used for the time discretization. The pressure Poisson equation enforces the incompressibility constraint for the velocity field, and the pressure is solved through the pressure Poisson equation with a Neumann boundary condition. We demonstrate by numerical results that this scheme is stable under the standard Courant–Friedrichs–Lewy (CFL) condition, and is second‐order accurate in time for the velocity, pressure, and divergence. Further, we develop three accurate, stable, and efficient solvers based on this algorithm by selecting different collocation points in r‐, ? ‐, and z‐directions. Additionally, we compare two sets of collocation points used to avoid the axis, and the numerical results indicate that using the Chebyshev Gauss–Radau points in radial direction to avoid the axis is more practical for solving our problem, and its main advantage is to save the CPU time compared with using the Chebyshev Gauss–Lobatto points in radial direction to avoid the axis. Copyright © 2010 John Wiley & Sons, Ltd.     

Spectral/hp penalty least‐squares finite element formulation for unsteady incompressible flows

In this paper, we present spectral/hp penalty least‐squares finite element formulation for the numerical solution of unsteady incompressible Navier–Stokes equations. Pressure is eliminated from Navier–Stokes equations using penalty method, and finite element model is developed in terms of velocity, vorticity and dilatation. High‐order element expansions are used to construct discrete form. Unlike other penalty finite element formulations, equal‐order Gauss integration is used for both viscous and penalty terms of the coefficient matrix. For time integration, space–time decoupled schemes are implemented. Second‐order accuracy of the time integration scheme is established using the method of manufactured solution. Numerical results are presented for impulsively started lid‐driven cavity flow at Reynolds number of 5000 and transient flow over a backward‐facing step. The effect of penalty parameter on the accuracy is investigated thoroughly in this paper and results are presented for a range of penalty parameter. Present formulation produces very accurate results for even very low penalty parameters (10–50). Copyright © 2008 John Wiley & Sons, Ltd.